Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $r \neq 0$. $x = \dfrac{-r + 8}{-r^2 + 11r - 24} \times \dfrac{-5r^2 - 35r - 60}{r + 4} $
Solution: First factor out any common factors. $x = \dfrac{-(r - 8)}{-(r^2 - 11r + 24)} \times \dfrac{-5(r^2 + 7r + 12)}{r + 4} $ Then factor the quadratic expressions. $x = \dfrac {-(r - 8)} {-(r - 8)(r - 3)} \times \dfrac {-5(r + 4)(r + 3)} {r + 4} $ Then multiply the two numerators and multiply the two denominators. $x = \dfrac {-(r - 8) \times -5(r + 4)(r + 3) } { -(r - 8)(r - 3) \times (r + 4)} $ $x = \dfrac {5(r + 4)(r + 3)(r - 8)} {-(r - 8)(r - 3)(r + 4)} $ Notice that $(r - 8)$ and $(r + 4)$ appear in both the numerator and denominator so we can cancel them. $x = \dfrac {5(r + 4)(r + 3)\cancel{(r - 8)}} {-\cancel{(r - 8)}(r - 3)(r + 4)} $ We are dividing by $r - 8$ , so $r - 8 \neq 0$ Therefore, $r \neq 8$ $x = \dfrac {5\cancel{(r + 4)}(r + 3)\cancel{(r - 8)}} {-\cancel{(r - 8)}(r - 3)\cancel{(r + 4)}} $ We are dividing by $r + 4$ , so $r + 4 \neq 0$ Therefore, $r \neq -4$ $x = \dfrac {5(r + 3)} {-(r - 3)} $ $ x = \dfrac{-5(r + 3)}{r - 3}; r \neq 8; r \neq -4 $